Talk:Elliptic curve

2003 material

I'm confused here. In the text, it states:

By adding a point "at infinity", we obtain the projective version of this curve; every straight line intersects this curve in three points (if the line is tangent to the curve at a point, then that point is counted twice). It is then possible to introduce a group operation on the curve with the following property: if a straight line intersects the curve at the points P, Q and R, then P + Q + R = 0 in the group.

But, for example, the x-axis intersects the curve y2 = x3 - x + 1 at only 1 point (call it P); if we consider the other two points as being infinity, this seems to require that P + inf + inf = 0. However, the x-axis is not alone in this property; several lines parallel to the x-axis also intersect at only 1 point, if we select one of these and call the point of intersection Q, then Q + inf + inf = P + inf + inf = 0 implies P = Q. Bzzzt!!!! Is the group operation restricted to those lines which actually intersect at at least two points (where tangency counts as 2 points)? Chas zzz brown 02:00 Jan 22, 2003 (UTC)

My statement above was wrong: it only that way only for algebraically closed base fields. You're right: if the fields isn't algebraically closed, we only consider lines that are tangent to the curve or intersect it in two points. AxelBoldt 17:23 Jan 22, 2003 (UTC)

I'd like to see a bit more about the group aspect of elliptic curves over finite fields (with an eye towards ECC); especially more describing an algebraic (as opposed to geometric) approach to calculating P + Q (see [1] for a nice explanation). Should that be done at this article, at the article for elliptic curve cryptography, or at a new article? Chas zzz brown 20:04 Feb 7, 2003 (UTC)

I think it can still fit here. Can't we give the general group law formula which works for all base fields? AxelBoldt 07:37 Feb 9, 2003 (UTC)

Perhaps the best way to handle this would be to emphasise that the geometric picture is only valid for the real field, and then to explain that nevertheless we can make sense of things over any field by working out some algebraic formulation. The really important point here is that if two given points have rational coordinates (i.e. belong to some field), then so does the third point. This is why the group operation makes sense when defined over a given field. --Dmharvey 21:00, 28 May 2005 (UTC)

Added the restriction that K not have characteristic 3 - observe that in the diagrams in the article, there is a point P with the property P + P + P = 3P = 0 (in the y2 = x3 - x - 1 example, it's the intersection of the y-axis and the curve). Chas zzz brown 19:57 Feb 12, 2003 (UTC)

Page needs sections now. Charles Matthews 09:51, 21 Mar 2005 (UTC)

nother picture

Sean κ. + 23:33, 27 May 2005 (UTC)

graphs of singularities

Would be nice to have graphs of singular weierstrass equations, illustrating both cusps and intersections.

On this point, it would also be nice to mention that the group law can still be defined for a singular curve, as long as we leave out the singular point (of which there can only be one -- I think). Dmharvey 17:00, 30 May 2005 (UTC)

Hi! There's one here:
http://en.wikipedia.org/wiki/Cubic_curve

Form of the polynomial

The article describes elliptic curves as being of the form ${\displaystyle y^{2}=x^{3}+ax+b}$. However, in Dan Bernstein's Elliptic Curve Diffie Hellman library, he uses the curve ${\displaystyle y^{2}=x^{3}+Ax^{2}+x}$ where ${\displaystyle A=486662}$. Is this still an elliptic curve?

If you think about the substitution x = x′ + c you see that there is some freedom here. That is, provided you can divide by 3 in the field, you can set c so that the Bernstein form has the coefficient of x′2 changed to 0. Charles Matthews 15:58, 26 October 2005 (UTC)

History of the term

Does anyone know why these curves are called elliptic curves? My apologies if I missed it in the article. -Monguin61 01:47, 10 December 2005 (UTC)

The connection is: elliptic curve => Weierstrass elliptic function (parameterises an elliptic curve) => elliptic integral => arc length of an ellipse. I tend to think of the name "elliptic curve" as a historical accident, and a poor name for the object it describes. Dmharvey 01:54, 10 December 2005 (UTC)

Rules for adding points

The article covers four cases for adding points, but it leaves out the case P+P where P is a point of inflection of the curve such that the line is horizontal or otherwise fails to intersect the curve at a third finite point. I don't want to edit the article because I'm far from an expert on this topic, but this is a fundamentally different case, is it not? --24.15.231.4 04:56, 16 December 2005 (UTC)

So in this case, the tangent line intersects the curve with multiplicity 3, so you get P + P + P = 0. (This includes 0 + 0 + 0 = 0.) I think you're right, the article deserves an extra diagram illustrating this, or at least some text describing it. Dmharvey 13:11, 16 December 2005 (UTC)
Say the tangent line is horizontal. Then P + P must be the point in infinity. Else, it must intersect the curve. I think that, given the curve
E: y^2 = x^3 + ax^2 + bx + c,
the formula of P + P is
((x^4 - 2bx^2 - 8cx + b^2 - 4ac)/(4y^2), (x^6 + 2ax^5 + 5bx^4 + 20cx^3 + (-5b^2 + 20ac)x^2 + (-2ab^2 + 8a^2 c - 4bc)x + 4abc - 8 c^2 - b^3)/(8y^3)),
unless of course, y = 0, where 2P is the point (0,1,0) in infinity.
I'm not sure exactly what you are saying here, but I can give an example. Take the curve ${\displaystyle y^{2}=x^{3}-432}$. This has a point of inflection at ${\displaystyle P=(12,36)}$, which can be checked by basic calculus. The slope of the curve at P is 6, so the tangent line has equation ${\displaystyle y=6x-36}$. Solving this simultaneously with ${\displaystyle y^{2}=x^{3}-432}$ gives the equation ${\displaystyle (x-12)^{3}=0}$ (as it should, because it's a point of inflection!). In other words, the tanget line intersects the curve only at P, with multiplicity 3. So in this case P + P + P = 0. In fact there should be altogether 9 points satisfying Q + Q + Q = 0; the point at infinity, P and -P, and six others with complex coordinates. Dmharvey 12:27, 25 March 2006 (UTC)

In the article they cite 'second pane' for case 'yP = yQ' whereas in pane 2 it is obviously not the case, seems to me ?Cédric VAN ROMPAY (talk) 14:44, 12 November 2013 (UTC)

Non-singular cubic curves?

What do we call those? I've heard them being called Elliptic curves. Or non-singular curves on this form

y^2 z + axyz + byz^2 = x^3 + cx^2 z + d x z^2 + e z^3

being called elliptic curves...

Your definition appears to be different.

Actually, those ARE elliptic curves. But every elliptic curve on a field with Characteristic different than two or three is isomorphic (in some Morphisms that preserve the Group Structure of the curve) to a curve like the ones defined in the article. So by giving the other definition, we didn't miss any information. BrunoX 02:03, 7 December 2006 (UTC)

DAB help needed

Can someone with knowledge disambig varieties in the Isogeny section? Thanks. Simon12 02:26, 1 July 2006 (UTC)

I took my best guess. Change if incorrect. Simon12 03:15, 1 July 2006 (UTC)

a Mistake?

"By adding a "point at infinity", we obtain the projective version of this curve"

why one point? by Bezout theorem, this curve has exactly 3 points at infinity. 84.108.112.10 12:50, 14 April 2007 (UTC)

Doesn't this statement depend on the embadding of the elliptic curve? Liransh 19:54, 3 May 2007 (UTC)

Remember, in Bezout's theorem intersection points are counted with multiplicities. Homogenizing the normal form equation of the article gives ${\displaystyle Y^{2}Z=X^{3}+aXZ^{2}+bZ^{3}}$. Set Y=1 and you'll find precisely one intersection of the curve and the line at infinity Z=0. And you'll easily notice that this is a triple intersection (look at the equation in the affine piece with ${\displaystyle Y\neq 0}$). Just to verify, if Y=0 and Z=0 you find no points. Also from the point of view of the group law this is what one should expect: the point at infinity is the neutral element of the group law, which requires it to be an inflection point for the geometric description of the group law to work.
And finally, yes, all this depends on the embedding. If you change that, you'll change the base point that acts as the neutral element of the group law. One way to think about the group law is that the Jacobian variety of the curve has intrinsically a group structure (not just for elliptic curves but in general). Choosing a base point on the curve gives you the corresponding Abel-Jacobi map from the curve to the Jacobian; the base point lands at the neutral element of the Jacobian. That is bijective for elliptic curves, and the group law of the curve can be defined by transporting the Jacobian's group law to the curve via this bijection. Thus changing the base point gives you a different neutral element on the curve. Stca74 14:20, 15 May 2007 (UTC)

Article rating

• Field: Clearly geometry, not algebra. Otherwise all of algebraic geometry would be under algebra, which does not make sense. And the transcendental theory of elliptic curves (over complex numbers) is even more clearly not algebra.
• Importance: High — these objects are among the most studied in 20th and 21st century pure mathematics and have an unusually large number of links to other fields. Individual (types of) objects should not typically be of HIGH importance, but elliptic curver in my opinion deserve that.

Stca74 12:53, 26 May 2007 (UTC)

Hello, as I explained on project talk page, these things tend to be both contentious and subjective. In this instance, though, I would explain why we disagree. First, elliptic curves combine not two, but THREE fields: algebraic geometry, complex analysis, number theory. Your two arguments are fine in isolation, but contradict each other: the transcendental theory of elliptic curves was finished in 19th century. Most if not all of recent research is in arithmetic, clearly not in geometry. Besides, I argued before that algebraic geometry should be a separate field, but the consensus seemed to be to leave the field alone and judge importance on the basis of categories. The reductio ad absurdum type arguments are not helpful, because clearly, we are dealing with an intentionally limited classification scheme, so contradictions are unavoidable (to me, 'Geometry'='Differential geometry', not 'Algebraic geometry'). Rather than ridiculing the classification, you can try to convince other parties why you believe it should be made more flexible. Best, Arcfrk 13:13, 26 May 2007 (UTC)
P.S. I agree with 'High' importance, though! It's hard to maintain stable criteria while going over dozens of articles, but thinking over it again, it's high, not mid.Arcfrk 13:35, 26 May 2007 (UTC)
Yes, I noticed your comment right after making the change here. And I thought about the number-theoretic importance of elliptic curves before posting my comment (that's the direction I'm coming from as well). And I think I would tend to support the separation of algebraic geometry from Geometry and Topology in the field classification. However, for as long as we do not have AG as its own field, I still think it (all of it) belongs under Geometry and Topology. The reason is that what really separates (in my mind, at least) AG from algebra is the essentially geometric viewpoint, valid also in surpisingly many arithmetic and positive-characteristic situations. And further support in my mind is provided by the essential cohomological methods which provide an extremely strong link to the topological side of Geometry and Topology. Finally, I think that from the viewpoint of a possible eventual splitting of the field, it will be easier if all of AG is for now withing one field (and again, classifying all of it with algebra would not work, as there is still a lot that is fully geometrically motivated with no arithmetic links. Cheers, Stca74 14:40, 26 May 2007 (UTC)
I have to admit I am somewhat perplexed by an argument that because elliptic curves are important in number theory, they should be rated within algebra. I am also puzzled why maths ratings have got anything to do with what are the topics of current research.
Geometry means a surprisingly large range of things, as a quick glance at Category:Geometry will soon reveal; it is certainly not equal to differential geometry for WP. Geometry guy 18:09, 30 May 2007 (UTC)
It only appears puzzling because it was a hypothetical discussion (if you think the transcendental theory is what is most important then you cannot claim at the same time that it's the current research that determines the importance, because the current research is in number theory). My position, hopefully, easy enough to understand, is that algebraic geometry should be a separate field. In the absence of agreement on that, elliptic curves should (in my opinion) better be filed under Algebra. However, I do not think that this in itself is of sufficient importance to spend any more time on discussion, whichever field and/or importance ratings have been assigned. Rather, we know the article is here, is fairly important, and awaits improvements. Arcfrk 05:59, 31 May 2007 (UTC)
"Here here" to the last sentence! Okay, you have cleared up my puzzlement, thank you for that, but not my perplexity: in the absense of field=AlgGeom, why do you prefer field=Algebra to field=Number theory? Geometry guy 18:36, 31 May 2007 (UTC)

I am an arithmetic geometer working in the field(s?) of elliptic curves and curves of genus one. Here are my reactions to the article as currently written (it is quite good overall):

The basic definition -- "an elliptic curve is..." is close but needs even more care: first of all, if one is going to define it in earnest using the language of algebraic / arithmetic geometry (BTW, it is not clear to me that one should do this right away; more people would understand the y^2 = x^3+ax+b definition, which is not wrong, just overly concrete for some purposes), you are missing the proviso "geometrically connected." More seriously, you are missing a mention of the field of definition of the elliptic curve. As stated above, for about 100 years it has been the case that elliptic curves are interesting only over non-algebraically closed fields, so the dependence on the field should be made clear from the start. Again, a reasonable choice would be to say a bit first about elliptic curves over the rational field and then start again with a definition over an arbitrary field (with the longer Weierstrass equation).

Nitpicking point: the definition of an elliptic curve in short Weierstrass form is not insufficiently "precise"; it is simply not the definition of an elliptic curve in char. 2 or 3 that agrees with the above geometric definition (i.e., the problem is not precision; it's simply a matter of a statement which is no longer true).

The paragraph beginning "If y^2 = P(x)..." is a bit awkwardly written. In particular, somewhere in this paragraph the fact that an elliptic curve must have a rational point is forgotten. The point of the Weierstrass form is that it comes equipped with a canonical rational point (the inflection point at infinity). The other forms -- hyperelliptic quartic, plane cubic and intersection of two space quadrics -- are important alternate ways of presenting smooth curves of genus one, but in these other forms a rational point is not guaranteed (rather, a rational divisor of degree 2, 3, or 4, respectively).

Saying that the proof of FLT is by Wiles assisted by Taylor is not ideal -- the work of Taylor and Wiles was indeed spectacular, but it also built on important work of many other mathematicians: e.g. Ribet, who proved that Taniyama-Shimura implies FLT. The attribution of FLT is a complicated story which can be left to the article itself; no need to try to wing it here.

"By adding a 'point at infinity', we obtain the projective version of this curve" This could be better phrased. The projectivization of an affine curve is an easy general construction: we just homogenize the defining polynomial by inserting in each monomial term whatever power of Z brings the total degree of each term up to the maximum of the total degrees of the original terms (in this case 3, of course). Surely this is described somewhere onsite? (If not, it should be!) Then the fact that the line at infinity intersects a Weierstrass cubic in a single point can be verified. My point is that the way it is said it sounds like one knew in advance that there was one point to add. This is really not the case, since e.g. one could work with a non-Weierstrass cubic endowed with a rational point, and then the projectivization could add as many as three points.

In your definition of the group law: you never actually say what P+Q _is_, you just give defining properties. Why not say that if R is the third intersection point of P and Q with the curve, then P+Q is the third intersection point of O and R with the curve (and illustrate this with a picture)?

"One can check that this turns the curve into an abelian group, and thus into an abelian variety." Bad -- an abelian variety is a commutative group variety but not conversely (think of the additive group or the multiplicative group, e.g.). The sentence could read, "One can check that the K-rational points on E form an abelian group under +. [insert some indication that the associativity is the sticky point!] Moreover, as a map from E x E to E, the addition law is "algebraic", i.e., given by rational functions on E, so endows E with the structure of an algebraic group (and indeed, since E is projective and geometrically connected, an abelian variety." Or you could not say all these things -- it's not really necessary.

"The above group law can be described algebraically as well as geometrically." I object to this dichotomy here and elsewhere in the article: the above description of the group law is both algebraic _and_ geometric (and also arithmetic, since it takes fields of rationality into account). It is not necessary to perform the construction first over the real numbers and then describe it separately -- it makes sense over any field using simple, purely algebraic definitions of partial derivative, tangent line, etc. "Algebraically" and "using the following explicit equations" are not synonymous!

Elliptic curves over the complex numbers: "curious property of Weierstrass's elliptic functions..." "Curious" is unencyclopedic. Anyway, it is not explained. "Looks like" |-> "is homeomorphic to" (with a link). The reader who has made it this far will not be confused. ... "then the corresponding elliptic curves are isomorphic" And conversely!

"The isomorphism classes can be understood in a simpler way..." "Simpler" is POV.

"The complex numbers are the splitting field for polynomials" -> "the complex numbers are algebraically closed"

Drop the name "Legendre form" for the y^2 = x(x-1)(x-lambda) and give a reference (say, to Silverman) for all these formulas.

"The uniformization theorem states" |-> "The uniformization theorem implies." (It says other things as well.)

Elliptic curves over a general field: again, I wish this would be more gracefully incorporated into the above. (One way to do it is to do it over Q in a way that works over an arbitrary field, and then later to announce this.)

"One typically takes the curve to be..." This is not quite right; a curve is a curve, in the sense of arithmetic geometry. It is not simply the set of its points over the algebraic closure (although the set of points together with all morphisms from it are sufficient to recover the algebraic structure). It is better not to say anything about this subtle point than to get it a little bit wrong.

Isogeny of elliptic curve: the term "basepoint" (should be "base point") has not been used above. (Better, e.g. "the origin", or "the neutral element".) About the homomorphism property of an isogeny: moreover, the homomorphism is surjective over the algebraic closure.

"no general algorithm is known..." Elaborate on this point. There is an algorithm which we use in practice, which will work provided Shafarevich-Tate groups are finite. (The point is that having an "algorithm" that has not been proved to terminate in all cases is not the same as not having any idea at all how to start computing.)

"A formula for this rank is given by..." This is true, but the other terms in the formula are difficult both theoretically and computationally as well.

"This fact can be understood and proven with the help of some general theory..." Well, okay. I wonder whether the people who are conversant (or even potentially conversant) with etale cohomology are really looking on Wikipedia to find out how to prove Hasse's bound. Of course Hasse's proof was simpler, and there is also a relatively simple proof in Silverman's book: a reference here would be nice.

"The number of points on a specific curve..." It would be hard to compute the number of points on a nonspecific curve, wouldn't it? :)

It would be nice also to have a section on elliptic curves over local fields and reduction theory, as well as a little bit about complex multiplication.

Well, I can help out with these changes, but not tonight. 128.192.134.50 06:58, 14 August 2007 (UTC)Plclark

Why does an elliptic curve have to have a rational point? If it doesn't, it's not called an elliptic curve?? Tkuvho (talk) 13:30, 2 December 2010 (UTC)

Associativity

Is it obvious to everybody but me that the elliptic-curve "addition" operation is associative? How on earth did anybody ever notice, "Hey, if I define a combination operation like this, it's associative, so I can get a group!" 68.189.88.169 (talk) 22:42, 2 March 2008 (UTC)

It's not obvious to anyone. However, the definition of addition on elliptic curves is quite natural, if you are familiar with Bezout's theorem, and once you have a commutative binary operation with identity and inverses it's not hard to conjecture that it constitutes a group operation, and then the proof writes itself (with many computations). The other perspective, that the points of an elliptic curve correspond to divisors of degree 0 in its Picard group, is not so obvious, but once you see it, you immediately get a group operation on the points and you start to wonder whether the other operation which you couldn't prove associative actually is, and is the same operation. And from the definition of the Picard group it is quite easy to prove they are the same thing.
Neither of these qualifies as "obvious", but they are not obvious in different ways. By the way, I disclaim the usefulness of either of the links I made; the Picard group page is really utterly unhelpful.
Finally, you might do better by asking this question in a real mathematical forum, which this isn't. Ryan Reich (talk) 23:10, 2 March 2008 (UTC)

The article now contains a very nice animation, and purports this to provide a geometric proof of associativity. I have seen proofs of associativity with very similar diagrams (in the textbook by Niven, Zuckerman, and Montgomery, for example), but the proof includes considerably more material than currently in this wikipedia article. For example, Bezout's theorem says two cubics intersect in nine points, but one may need even more to complete the proof of associativity: such that if three cubics share eight points, then the ninth point in each of pairwise intersections is identical. Just to be clear I am only guessing that something like that would complete the proof. (By the way, in the diagram the three cubics are (1) the curve, and (2) the three nearly vertical lines and (3) the three nearly horizontal lines. The ninth point in question is the central point.) In other words, the article is now wrong: a geometric proof is not supplied. I will try to remove this claim, without hindering the importance and intuition of the animation. I hope that somebody else can supply the missing parts of the proof. DRLB (talk) 15:50, 13 May 2013 (UTC)

First sentence

Did we lose something along the way? The O in the first sentence doesn't seem to help with the definition. Perhaps there was some characteristic of O that is missing? (John User:Jwy talk) 23:00, 19 October 2008 (UTC)

The requirement for a point O forces the curve to have a rational projective point. Curves without any rational projective points are not elliptic curves. In practice, most of the time, the curve is chosen to have a single point at infinity which is taken for O, but this is not a requirement. 83.159.7.181 (talk) 10:13, 17 July 2012 (UTC)

A typo?

I'm not an algebraic geometer/number theorist, but this looks wrong: "For prime numbers ℓ not dividing N, the coefficient \scriptstyle a(\ell) of the form equals ℓ – the number of solutions of the minimal equation of the curve modulo ℓ." Shouldn't it be "...equals a – the number of solutions..."? If so, probably needs a reference — Preceding unsigned comment added by 203.24.207.178 (talk) 06:10, 31 July 2012 (UTC)

Hey, Deltahedron...

why didn't you like my edit?

The link to Tunnell, as it stands, isn't useful. The referenced Tunnell isn't even one of those mentioned in the disambiguation page, whereas I proposed an useful link.

At least an explanation is in order instead of just bluntly undoing. — Preceding unsigned comment added by 89.247.124.39 (talk) 13:21, 24 December 2012 (UTC)

I don't understand this. Template:IPuser made this change to link to Tunnell's theorem and that link is still there. I assume that Template:IPuser is the same person? My subsequent change was quite unrelated. It would help if the anonymous editor(s) would (1) explain exactly what it is they are objecting to here, (2) address personal messages to other editors on User talk pages not article talk pages, and (3) consider acquiring their own editor name to facilitate communication with other users. Deltahedron (talk) 14:00, 24 December 2012 (UTC)

Original anonymous here: Oops -- I stand corrected. I can't reproduce the problem now: the new link is there. Let's assume it was my browser playing nasty caching tricks on me.

Please accept my apologies — Preceding unsigned comment added by 89.247.13.63 (talk) 10:20, 26 December 2012 (UTC)

Not a problem. But please do consider Wikipedia:Why create an account?. Deltahedron (talk) 17:28, 26 December 2012 (UTC)

What's E(Q)?

I came here looking to remind myself how the rational points of an elliptic curve were defined, and was annoyed that halfway through the article the symbols E(Q) is introduced without explanation. For lots of people this would mean nothing. I'm enough of a mathematician that I know it means the rational points of the elliptic curve... however, the article should explain this and say exactly what the rational points are. (For example, presumably the point at infinity is a rational point.) — Preceding unsigned comment added by John Baez (talkcontribs)